×

Sorry!, This page is not available for now to bookmark.

(i) Sine Function (sin)

sin \[\theta\] = \[\frac{Opposite}{Hypotenuse}\]

(ii) Cosine Function (cos)

cos \[\theta\] = \[\frac{Adjacent}{Hypotenuse}\]

(iii) Tangent Function (tan)

tan (\[\theta\]) = \[\frac{sin (\theta) }{cos (\theta) }\]

As stated above, the angles other than the 90 degrees angle in a right-angled triangle are acute (i.e, less than 90 degrees). To find the value of functions for angles more than 90 degrees, we follow a set of rules known as cast rule. Let us take four axes, to divide 360 degrees into four quadrants. The angles are always measured anti-clockwise from the positive x-axis. The given diagram shows 30 degrees from the positive x-axis.

[Image will be uploaded soon]

There are four quadrants as shown in the figure below. Each quadrant contains a range of angles:

First Quadrant: All the angles between 0⁰ and 90⁰ lie in the first quadrant. The value of all the functions (Sin, Tan, Cos) in this quadrant are positive. Shown as A (represents All) in the second diagram given below.

Second Quadrant: All the angles between 90⁰ and 180⁰ lie in the second quadrant. The value of only Sin is positive in this quadrant. Shown as S (represents Sin) in the second diagram given below.

Third Quadrant: All the angles between 180⁰ and 270⁰ lie in the third quadrant. The value of only Tan is positive in this quadrant. Shown as T (represents Tan) in the second diagram given below.

Fourth Quadrant: All the angles between 270⁰ and 360⁰ lie in the fourth quadrant. The value of only Cos is positive in this quadrant. Shown as C (represents Cos) in the second diagram given below.

[Image will be uploaded soon]

Starting from the fourth quadrant we can say that quadrant follows the series of CAST where C is for Cos, A for all, S stands for Sin, and T is for Tan.

The value of sin, cos, and tan of some angles are equal to the values of sin, cos, and tan of other angles. Let us take an example of cos(-30⁰). Since 30 degrees lie in the first quadrant we can say that cos(-30⁰) is equal to cos(30⁰) because all the angles in the first quadrant are positive. Similarly, cos(390⁰) also equal to cos(30⁰). The diagrams below show the shaded angle of sin, cos, and tan having the same magnitude.

Fig 1: sin 30⁰ = 0.5

Fig 2: sin 150⁰ = 0.5

Fig 3: sin 210⁰ = -0.5

Fig 4: sin 330⁰ = -0.5

[Image will be uploaded soon]

Therefore these angles are called related angles. The image given below shows the value of all angles of cos.

[Image will be uploaded soon]

The value of Cos 120⁰ is - ½. It is the additive inverse of the value of cosine 60⁰ or cos 60⁰. Cosine is one of the basic trigonometric functions. It is expressed as a ratio of the base of a right-angled triangle to its hypotenuse.

The angles of a right-angled triangle are expressed in terms of multiples or sub-multiples of 180⁰, or π in radians. Hence, to find the value of cos 120⁰, we will have to express 120⁰ in terms of 180⁰ or 90⁰.

Case 1:

Let us express 120⁰ as (180 - 60)⁰

cos 120⁰ = cos (180 - 60)⁰

Since, cos (180⁰ - x) = - cos x.

Therefore, cos (180 - 60)⁰ = - cos 60⁰

=> cos 120⁰ = -½

Case 2:

Let us express 120⁰ as (90 + 30)⁰

cos 120⁰ = cos (90 + 30)⁰

Since, cos (90⁰ + x) = - sin x.

Therefore, cos (90 + 30)⁰ = -sin 30⁰

=> cos 120⁰= - ½

Question 1: Find the Exact Value of cos (- 390⁰).

Solution: We know cos(-x) = cos(x)

therefore , cos (- 390⁰) = cos (390⁰).

Since the angle, 390⁰ is greater than angle 360⁰,

we find an angle t such that,

t = 390 - (360) = 30⁰

This means Cos of cos (390⁰) and cos ( 30⁰) coterminal.

So,

Cos (390⁰) = Cos ( 30⁰)

We know that the value of Cos 30⁰ is equal to \[\sqrt{\frac{3}{2}}\]

FAQ (Frequently Asked Questions)

Question: Why do We Use sin theta and cos theta in an Angle?

Answer: Sin theta and Cos theta explains an interesting relation between the sides of the right-angled triangle. Pythagoras theorem can also be used to find the length of the missing side of a right-angled triangle if the other two sides are known.

Functions like Sin, Cos, and Tan are also used to find the side of a right-angled triangle if one side and the angle is known. For example, a right angle triangle has 60 degrees as theta and hypotenuse is 10 cm.

Image will be uploaded soon

We know, Cos = Adjacent /Hypotenuse

According to the table of values of trigonometric ratios, Cos (60) = ½ .

So, placing the value of Hypotenuse,

Cos 60 = Adjacent /10 cm

1/2 = Adjacent /10 cm

Adjacent = 10 / 2 = 5 cm.